
Firstly, inspired by [1]}, by upper bounding arbitrary convex functions, which link the expected and empirical losses at environment and also per-task levels, we propose the general
PAC-Bayes meta-generalization bounds (Section ).

Proper choices of the convex functions recover known PAC-Bayes bounds including classic, quadratic and fast-rate families (Section ).

We provide a new fast-rate bound and also a new classic bound with better performance on the meta-test set and with convergence rate \(O(\sqrt{(1/+1/\mathrm {M})})\)  (Section ).
Following the meta-learning by adjusting the priors (MLAP) algorithm [2]}, we develop the MLAP algorithm for our new obtained bounds in the Section
.
We demonstrate the usefulness of the proposed bounds in an example in Section .
The main merit of our new classic bound is its significant performance to avoid meta overfitting.

